Adapter Signatures and Their Application in Cross-Chain Atomic Swaps

With the rapid development of Bitcoin Layer 2 scaling solutions, the frequency of cross-chain asset transfers between Bitcoin and its Layer 2 networks has significantly increased. This trend is driven by the higher scalability, lower transaction fees, and high throughput offered by Layer 2 technology. These advancements facilitate more efficient and cost-effective transactions, thereby promoting broader adoption and integration of Bitcoin across various applications. Therefore, interoperability between Bitcoin and Layer 2 networks is becoming a key component of the cryptocurrency ecosystem, driving innovation and providing users with a more diverse and powerful range of financial tools.

There are three typical solutions for cross-chain transactions between Bitcoin and Layer 2, namely centralized cross-chain transactions, BitVM cross-chain bridge, and cross-chain atomic swaps. These three technologies differ in terms of trust assumptions, security, convenience, transaction limits, etc., and can meet different application needs.

The advantages of centralized cross-chain trading lie in its speed and the relatively easy matching process, as centralized institutions can quickly confirm and process transactions. However, the security of this method completely relies on the reliability and reputation of the centralized institution. If the centralized institution encounters technical failures, malicious attacks, or defaults, users' funds are at a higher risk. Moreover, centralized cross-chain trading may also leak user privacy, requiring users to consider this method carefully.

The BitVM cross-chain bridge technology is relatively complex. This technology introduces an optimistic challenge mechanism, making it relatively complicated. Additionally, the optimistic challenge mechanism involves a large number of challenge and response transactions, resulting in higher transaction fees. Therefore, the BitVM cross-chain bridge is only suitable for very large transactions, similar to the issuance of U, thus its frequency of use is relatively low.

Cross-chain atomic swaps are a type of contract that enables decentralized cryptocurrency trading. Atomic swaps must involve two parties, and no third party can interrupt or interfere with the swap process. This means that the technology is decentralized, censorship-resistant, offers good privacy protection, and allows for high-frequency cross-chain transactions, making it widely used in decentralized exchanges.

The cross-chain atomic swap technology mainly includes hash time lock and adapter signature. The cross-chain atomic swap based on hash time lock (HTLC) has privacy leakage issues. The cross-chain atomic swap based on adapter signature has three advantages: First, the adapter signature swap scheme replaces the "secret hash" swap that relies on on-chain scripts, including time locks and hash locks. Second, since such scripts are not involved, the on-chain space occupation is reduced, making the adapter signature-based atomic swap lighter and cheaper. Finally, the transactions involved in the adapter signature atomic swap cannot be linked, achieving privacy protection.

Schnorr/ECDSA adapter signatures have issues with random number leakage and reuse, which need to be mitigated using RFC 6979. RFC 6979 specifies a method for generating deterministic digital signatures using DSA and ECDSA, addressing the security issues associated with generating the random value k.

In cross-chain scenarios, it is necessary to consider the heterogeneity problem between UTXO and account model systems. Bitcoin adopts the UTXO model and implements native ECDSA signatures based on the Secp256k1 curve. Bitlayer, being an EVM-compatible Bitcoin L2 chain, also utilizes the Secp256k1 curve and supports native ECDSA signatures. The adapter signature implements the logic required for BTC exchanges, while the counterpart for Bitlayer exchanges is supported by the powerful capabilities of Ethereum smart contracts.

If Bitcoin and Bitlayer both use the Secp256k1 curve, but Bitcoin uses Schnorr signatures while Bitlayer uses ECDSA, then the adapter signatures based on Schnorr and ECDSA are provably secure. However, if Bitcoin uses the Secp256k1 curve and ECDSA signatures, while Bitlayer uses the ed25519 curve and Schnorr signatures, then adapter signatures cannot be used.

Adapter signatures can also be applied to non-interactive digital asset custody. This method has the advantage of non-interaction, where the custodian cannot sign arbitrary transactions but only sends a secret to one of the supported parties. The implementation process requires the use of verifiable encryption technology, and currently, there are two promising methods, Purify and Juggling, which use verifiable encryption based on the Secp256k1 discrete logarithm.

In summary, adapter signatures have important applications in areas such as cross-chain atomic swaps and digital asset custody, but multiple aspects such as random number security and system heterogeneity need to be considered in practical use.